Cybernetic Interretation of the Riemann Zeta Function Petr Klán, Det. of System Analysis, University of Economics in Prague, Czech Reublic, etr.klan@vse.cz arxiv:602.05507v [cs.sy] 2 Feb 206 Abstract: This aer uses cybernetic aroach to study behavior of the Riemann zeta function. It is based on the elementary cybernetic concets like feedback, transfer functions, time delays, PI (Proortional Integral) controllers or FOPDT (First Order Plus Dead Time) models, resectively. An unusual dynamic interretation of the Riemann zeta function is obtained. Key words: Riemann zeta function, feedback, time delay, PI controller, FOPDT model Introduction Zeta function [4] is associated with equation n= n = s =2,3,5... s () introducing the fundamental relationshi between natural numbers n =,2,3,4,5... and rime numbers = 2,3,5,7,... for real s >. It was discovered by L. Euler in 737. A full idea of this relationshi excels at exansion to the individual base elements + s 2 + s 3 + s 4 + s 5 +... =..., s 2 s 3 s 5 s 7 s s in which an infinite summation over all the natural numbers (left side) and an infinite roduct over all the rime numbers (right side) are contained. L. Euler assumed the arameter s on both sides as a real number. Equation () is considered to be an inut to the mystery of interrelation between natural and rime numbers that is waiting for hundreds of years to be discovered [2]. The infinite sum /n s from (), tyically for s >, is called zeta function ζ(s), thus ζ(s) = n= ns. (2) Function diverges for s < and converges for s >. For s =, the function ζ(s) is rojected to a harmonic series, which is divergent as well [5]. G. F. Riemann rooses a secific analytic function in the field of all comlex numbers s = a + ib in his work On the Number of Prime Numbers Under a Given Magnitude ([4], Riemann s original work is a art of this ublication) that coincides with zeta function for Re(s) >. This new function is called the Riemann zeta function and denoted in the same way aszeta function(2). Whilst coincident with(2)introduced byl.euler forvaluesofa > and b = 0, it is more formally considered for values of a <. The Riemann zeta function
stays in the effect throughout the interval of s, including a <. In considering language of the comlex analysis, the Riemann zeta function reresents a meromorhic function in whole comlex lane excet oint s =, where it has no final value [8]. It greatly contributes to the analysis of rime numbers distribution [2, 4, 9]. Riemann zeta function ζ(s) has two sorts of zero values 2 or values s for which ζ(s) = 0. Some of them, for examle s = 0.5+i4.35, s = 0.5+i2.022, s = 0.5+i25.0, s = 0.5+i30.425, s = 0.5+i32.935... are called nontrivial. One can ask, where do all the comlex zero values of the Riemann zeta function locate. As extremely interesting, it is taken the area of comlex lane determined by interval 0 a for any real b which is called critical stri, see Fig.. The Riemann zeta function has zero values in negative integers ζ( 2) = 0, ζ( 4) = 0, ζ( 6) = 0...as well. They are called trivial. The Riemann hyothesis conjectures that all the critical stri nontrivial zeros lie on midst critical line of /2. In other words, all the zero values of the Riemann zeta function (excet trivial) are assumed to receive values of s = 0.5+ib or /2+ib for real b. When finding a single zero value in the critical stri outside this midst line, Riemann hyothesis fails. The hyothesis has not been roven yet. Due to its fundamental significance, it belongs to the millennium roblems [2]. 40 30 20 0 40.9 37.6 32.9 30.4 25.0 2.0 4. Critical line 0 /2 0 4. 20 2.0 25.0 30 30.4 32.9 40 37.6 40.9 Critical stri Figure : Critical stri of comlex lane showing ositions of nontrivial zero values. The aer shows that for a >, the Riemann zeta function can be interreted by an endless serial connection of the elementary cybernetic concets like PI controllers or FOPDT models [6]. It is shown that this interretation brings a new ersective to the Riemann zeta function with ossible imacts to the Riemann hyothesis. This article rovides an elegant roof of equation () for Re(s) >. 2 Terms like zeros and oles of transfer functions are well known in cybernetics too [6]. 2
2 Single transfer functions The right hand side of equation () is comosed of an infinite roduct of base elements s, (3) where is a rime and s a comlex number. Cybernetic considerations frequently use the base of natural logarithm e unlike rimes in the number theory. Thus, rewrite the base element (3) equivalently to e sln. (4) If e x = s for a certain x, then x = sln. Now, the infinite roduct (2) becomes e =... (5) sln e sln2 e sln3 e sln5 e sln7 e sln In Lalace transformation [3], the exression e sln reresents a transfer function including time delay ln. Generally, G(s) = e sl, where L = ln [6]. Looking through the eyes of transfer functions, a block scheme including ositive feedback for the entire exression (4) is in Fig. 2. Here, R(s) is Lalace transformation of a reference inut and Y(s) is Lalace transformation of the corresonding outut. In that case Y(s)/R(s) = /( e sln ). R(s) + Y(s) + e sl Figure 2: Base element (4) reresented by the transfer function. Reference inuts are usually realized by a unit ste (t). The latter is defined as 0 for times t < 0 and for t > 0, in Lalace transformation /s. Assume that r(t) = (t) is a unit ste and determine outut resonse y(t) of transfer function (4) to a unit ste where the signals r(t) and y(t) are denoted by R(s) and Y(s) in the Lalace transformation. Suose time delay L = ln 2 related to the first rime number in the ositive feedback loo of Fig. 2. Corresonding transfer function is Y(s)/R(s) = /( e sln2 ). A reference inut unit ste is immediately transferred to the outut, or y(t) = for t > 0. This outut is delayed in the feedback and after time delay L it is added to the reference inut, or y(t) = 2 for t > L. It again is delayed by time L, added to reference inut. It results in y(t) = 3 for t > 2L etc. A whole resonse to unit ste reference inut (transient resonse) is in Fig. 3. Due to the ositive feedback, a regular multiste signal y(t) is obtained 3. 3 It agrees with [3] which introduces the same Lalace transformation for time functions shown in Fig. 3. 3
y 8 7 6 5 4 3 2 0 ln2 3ln2 5ln2 7ln2 t Figure 3: Ste resonse associated with transfer function /( e sln2 ). Transient resonse in Fig. 3 can not be simly described. Therefore, an aroximation is used in the sequel. The oints laced just in the middle of time delays in Fig. 3 join a line beginning at y(0) = 0.5. Some of these oints and the corresonding values of transient resonse are collected in Tab.. t ln2/2 3ln2/2 5ln2/2 7ln2/2... y 2 3 4... Table : A collection of transient airs associated with transfer function /( e sln2 ). Considering airs in Tab. one can make the following interesting deduction: the aroximated line transient resonse is described by y(t) = 0.5+ ln2 t which coincides with the transient resonse of a PI controller given by transfer function 2 + sl or 0.5 + /sln2, where L = ln2 reresents an integration time constant. Notice that the line in Fig. 3 corresonds to (6) as well if the first order Pade aroximation is used in the form e sln = ( sln/2)/(+sln/2). After substituting the latter into /( e sln ) it imlies that /[ ( sln/2)/(+sln/2)] = /2+/(sln). (6) 4
Since an infinite roduct in the Riemann zeta function, it is concluded that the latter can be aroximated by an endless series combining transfer functions of PI controllers (6) ( 2 + ) (7) sln or ( 0.5+ )( 0.5+ )( 0.5+ )( 0.5+ )( 0.5+ )... sln2 sln3 sln5 sln7 sln This aroximation is called PI aroximation or simle aroximation of the Riemann zeta function. Connection of something fundamental as the Riemann zeta function is in the theory of numbers and something as PI controllers fundamental in cybernetics is really surrising based on a big context difference of both these subjects. 3 Case ζ(ib) By virtue of formulation (5) in which s = ib for non negative real b and a = 0, we can form ζ(ib) by e ibln. Based on (7), it results in a PI aroximation ζ PI (ib) = ( 2 + ) ibln comosed of the frequency transfer functions. 4 Case ζ(a+ib) Consider s = a+ib for non negative real a,b. Thus, each base element of the zeta function (4) equals and by rearranging e (a+ib)ln (8) e al e ibl, (9) in which L = ln. Introducing K = e al as form of a gain, it comoses a feedback loo with ositive feedback shown in Fig. 4. Analogously with Fig. 2, it relates to transfer function Y(s)/R(s) = /( Ke sl ) for s = ib. Consider again L = ln2 for time delay in Fig. 4. If a reference unit ste inut occurs, then it immediately is transferred to the outut, i.e. y(t) = for t > 0. In feedback, it is multilied by gain K, delayed by time L, and added to the reference inut, i.e. y(t) = +K for t > L. It again is multilied by K, delayed by time L, added to reference inut, which roducts 5
R(s) + Y(s) + e sl K Figure 4: Base element (9) as a feedback loo, s = ib. y(t) = +K(+K) = +K+K 2 for t > 2L, y(t) = +K(+K(+K)) = +K+K 2 +K 3 for t > 3L etc. It forms transient resonse which is shown in Fig. 5. Note that a secific case of this resonse for K = is in Fig. 3. A comlex ste signal y(t) is generated by an irregular height of individual stes. In Fig. 5, oint +K on vertical axis relates to +K, +2K to +K +K 2, +3K to +K +K +K 2 +K 3 etc. +7K +6K +5K +4K +3K +2K +K y 0 ln2 3ln2 5ln2 7ln2 t Figure 5: Transient resonse associated with transfer function /( Ke sln2 ). Since K and K > 0, summation +K+K 2 +...+K n constitutes geometric series having artial sums +K+K 2 +...+K n = ( K n+ )/( K) and the infinite sum /( K). A few single time oints in Fig. 5 together with related values of transient resonse is collected in Tab. 2. Since a close relation between geometric series and exonential functions, the artial sum of the geometric series can be written as ( ) K n+ = ( ) e (n+)lnk, (0) K K 6
n 2 3 4... K 2 K 3 K 4 K 5 y K K K K... Table 2: Values of transient resonse reresenting transfer function /( Ke sln2 ). in which n 0 and n = 0 relates to t = 0, n = to t = ln2, n = 2 to 2ln2 etc. It exresses transient resonse of the first order transfer function + Ke sl +Ts, () where K is gain, T time constant and L time delay. This kind of transfer function is called FOPDT model or three arameter model [6, 7]. Gain of transfer function () is given by the steady state of unit ste resonse K = /( K) owing to the initial condition y(0) =, then K = K K. (2) Identification of the time constant T is based on airs in Tab. 2 giving T = ln2 K. (3) It is based on scenario, in which the tangent line of transient resonse at beginning intersects the oint K indicated in Fig. 2. If the ratio of K/( K) and time constant T determines ẏ(0), then the latter is determined by the ratio K and ln2 as well. This aroximation suggests that L = 0 ermanently for all time delays of FOPDT model (0). A deeer analysis of transient resonses for large rime numbers associated with big delays L and small gains K shows that the shae of the resonse in Fig. 5 is relatively rare. More often, resonses are similar to single stes characterized by delay L and one magnitude +K since the very small squares of K (0). This observation was indeendently confirmed by a use of the three arameter model calculus [7], which underlines the imortance of the time delay aroximation in describing behavior of feedback loo in Fig. 4. Use of this calculus also showed that the time constant (3) estimates rather give the average residence time of FOPDT model T ar = T +L, see () than the time constants themselves. Thus, the estimated time constant results in and the estimated time delay T = ln2 K (4) K L = ln2 ( ) K. (5) K These estimates agree with observations. If K is very small, then the estimated time constant is tiny. Conversely, the time delay rises u. Resonse (0) looks almost like a single ste change in this case. Therefore, one can conclude that the Riemann zeta function ζ(a + ib) is described by the endless transfer function called FOPDT aroximation or comlex aroximation in the form ( ) + K e sl, (6) +T s 7
where and or by the FOPDT frequency function (6) K = e aln = a, K = K /( K ), T = ln K K L = ln K ( K ) ζ FOPDT (a+ib) = ( ) + K e ibl. (7) +ibt Note for comleteness, that for Pade aroximation of (9) we obtain /[ K( SL/2)/(+ sl/2)] = ( + sl/2)/[( K) + ( + K)sL/2], which eventually leads to transfer function based on the single base elements K +sln/2 ( ) +K + ln/2 K This aroximation indicates a different initial oint for the transient resonse. Whilst y(0) = for (6), y(0) = /( + K ) at Pade aroximation. If a = 0, then y(0) = 0.5 in accordance with (7). However, the Pade aroximation is only alicable to the base elements having small time delays. 5 FOPDT aroximation exeriments Some tyical real values (b = 0) obtained by FOPDT aroximation (6) are collected in Tab. 3. Exeriments were carried out with the first million of rime numbers []. a ζ(a) Tabular value a ζ(a) Tabular value 2.6449 π 2 /6 7.0083 3.202 8.004 π 8 /9450 4.0823 π 4 /90 9.0020 5.0369 0.000 π 0 /93555 6.073 π 6 /945 2.0002 69π 2 /63852875 Table 3: Some real values of the Riemann zeta function obtained through FOPDT aroximation.. s Results for comlex values of s = 2+bi obtained by FOPDT aroximation (7) summarize Tab. 4. Exeriments were carried out with the first million rimes as well. For comarison, Tab. 4 refers to tabular values of ζ(2+ib) on the first 4 decimal laces. The tables 3 and 4 show that real and comlex values of the Riemann zeta function obtained by FOPDT aroximation(7) are rather accurate. In the case of comlex values surrisingly u to two decimal laces since achieved by simle FOPDT model. 8
b ζ(2 + ib) Tabular value b ζ(2 + ib) Tabular value 0..635 0.0928i.6350 0.0927i 0.6.3807 0.404i.382 0.4037i 0.2.6067 0.799i.6067 0.798i 0.7.365 0.4262i.387 0.426i 0.3.5627 0.2567i.5628 0.2565i 0.8.2552 0.4375i.2582 0.4378i 0.4.5075 0.3202i.5079 0.398i 0.9.980 0.4399i.208 0.440i 0.5.4455 0.3692i.4463 0.3688i.0.459 0.4353i.504 0.4375i Table 4: Some comlex values of the zeta function obtained through FOPDT aroximation. 6 Conclusion Based on the similarity of transient characteristics, two aroximations of the Riemann zeta function for a >, ζ PI (see (8)) and ζ FOPDT (see (7)), were obtained. Aroximation of the Riemann zeta function results in a dynamic model consisting of base elements including ositive feedbacks and delays, which are infinitely many times serially connected. It rovides rather accurate aroximations owing to their simlicity. The aer oints to the ossibility of using the methods of cybernetics for solving roblems in number theory. Two cybernetic like models were derived close to the Riemann zeta function in terms of single transfer functions. Based on this aer it can be remarkably concluded, that such fundamental concets of cybernetics as PI controllers or FOPDT models stand in the modeling of the fundamental and comlex function as the zeta function undoubtedly is. Can cybernetics be used e.g. in the density analysis of rimes with resect to the known role of the Riemann zeta function in this subject? References [] Ch. Caldwell (994 202): The Prime Pages. Online rimes.utm.edu. [2] C.C. Clawson (996): Mathematical Mysteries. The Beauty and Magic of Numbers. Basic Books. [3] G. Doetsch (96): Anleitung zum Praktischen Gebrauch der Lalace Transformation. R. Oldenbourg, Műnchen. [4] H.M. Edwards (200): Riemann s Zeta Function. Dover Publications. [5] P. Klán (204): Numbers. Relationshis, Insights and Eternal Insirations. Academia Prague (in Czech). [6] P. Klán, R. Gorez (20): Process Control. FCC Public. [7] P. Klán (20): A Comrehensive Calculus of Three Parameter Models. Automa No. 8-9, 70 73 (in Czech). [8] M.Křížek, M.Markl, L.Somer(203): In 203, Pierre Delignewon the Abel Prize for fundamental work joining algebra and geometry. Pokroky matematiky, fyziky a astronomie, Vol. 58, No. 4, 265 273 (in Czech). [9] I. Stewart (203): Visions of Infinity. Basic Books. 9